Leonard pairs and the Askey- Wilson relations

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A : V → V and A : V → V which satisfy the following two properties: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V . Referring to the above Leonard pair, we show there exists a sequence of scalars β, γ, γ, ̺, ̺, ω, η, η taken from K such that both A 2 A ∗ − βAA ∗ A+ AA − γ (AA+AA)− ̺A = γA + ωA+ η I, A ∗2 A− βA ∗ AA + AA − γ(AA+AA)− ̺A = γA + ωA+ ηI. The sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey-Wilson relations.